Page 272 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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252 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
which freezing occurs. When a system is in equilibrium its
energy is in minimum level (dG = 0), which for a system Constant B is in fact same as H vap /R. Because of three ma-
t
jor simplifying assumptions made above, Eq. (6.101) is very
V
L
t
L
V
with only vapor and liquid is dG = d(n G + n G ) = 0. approximate and it may be used over a narrow temperature
V
L
Since dn =−dn , it can be written as [1]: range when minimum data are available. Constants A and
V
L
(6.96) dG (T, P sat ) = dG (T, P sat ) B can be determined from minimum two data points on the
vapor pressure curve. Usually the critical point (T c , P c ) and
where P sat indicates that the relation is valid at the saturation normal boiling point (1.01325 bar and T b ) are used to obtain
temperature and pressure. Similar equation applies to solid- the constants. If H vap is known, then only one data point (T b )
liquid or solid–vapor phases. would be sufficient to obtain the vapor pressure correlation.
During a phase change (i.e., vapor to liquid or vice versa), A more accurate vapor pressure correlation is the following
temperature and pressure of the system remain constant and three-constant correlation known as Antoine equation:
therefore from Eq. (6.5) we have:
B
sat
H vap (6.102) ln P = A −
(6.97) S vap = T + C
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
T
where H vap is heat of vaporization and S vap is the entropy A, B, and C, known as Antoine constants, have been deter-
of vaporization. H vap is defined as: mined for a large number of compounds. Antoine proposed
this simple modification of the Clasius–Clapeyron equation
H vap = H(T, P sat , saturated vapor) in 1888. Various modifications of this equation and other cor-
(6.98) − H(T, P sat , saturated liquid) relations for estimation of vapor pressure are discussed in the
next chapter.
Similarly S vap and V vap are defined. For a phase change
from solid to liquid instead of heat of vaporization H vap ,
heat of fusion or melting H fus is defined by the difference be- Example 6.6—For pure water, estimate vapor pressure of wa-
tween enthalpy of saturated liquid and solid. Since P sat is only ter at 151.84 C. What is its heat of vaporization? The actual
◦
a function of temperature, S vap and V vap are also functions values as given in the steam tables are 5 bar and 2101.6 kJ/kg,
of temperature only for any pure substance. H vap and V vap respectively [1]. Assume that only T b , T c , and P c are known.
decrease with increase in temperature and at the critical point
they approach zero as vapor and liquid phases become iden- Solution—From Table 2.1 for water we have T c = 647.3K,
tical. While V vap can be calculated from an equation of state P c = 220.55 bar, and T b = 100 C. Applying Eq. (6.101) at the
◦
as was discussed in Chapter 5, methods of calculation H vap
critical point and normal boiling point gives lnP c = A − B/T c
will be discussed in Chapter 7. By applying Eq. (6.8) to both and ln (1.01325) = A − B/T b . Simultaneous solution of these
L
V
dG and dG and use of Eqs. (6.97) and (6.98) the following equations gives the following relations to calculate A and B
relation known as Clapeyron equation can be derived: from T b , T c , and P c .
dP sat H vap
(6.99) = ln P c
dT T V vap 1.01325
B = 1 1
This equation is the basis of development of predictive meth- (6.103) T b − T c
ods for vapor pressure versus temperature. Now three sim- A = 0.013163 + B
plifying assumptions are made: (1) over a narrow range of T b
temperature, H vap is constant, (2) volume of liquid is small
in comparison with vapor volume ( V vap = V − V = V ), where T c and T b must be in kelvin and P c must be in bar.
V
V
L ∼
and (3) volume of vapor can be calculated from ideal gas law The same units must be used in Eq. (6.101). In cases that
(Eq. 5.14). These assumptions are not true in general but at a value of vapor pressure at one temperature is known it
a narrow range of temperature and low pressure conditions should be used instead of T c and P c so the resulting equa-
they can be used for simplicity. Upon application of assump- tion will be more accurate between that point and the boiling
tions 2 and 3, Eq. (6.99) can be written in the following form point. As the difference between temperatures of two refer-
known as Clausius–Clapeyron equation: ence points used to obtain constants in Eq. (6.101) reduces,
the accuracy of resulting equation for the vapor pressure be-
dln P sat H vap tween two reference temperatures increases. For water from
(6.100) =−
d(1/T) R Eq. (6.103), A = 12.7276 bar and B = 4745.66 bar · K. Substi-
tuting A and B in Eq. (6.101) at T = 151.84 + 273.15 = 425 K
where R is the universal gas constant. This equation is the
basis of development of simple correlations for estimation gives ln P = 1.5611 or P = 4.764 bar. Comparing predicted
of vapor pressure versus temperature or calculation of heat value with the actual value of 5 bar gives an error of −4.9%,
of vaporization from vapor pressure data. For example, by which is acceptable considering simple relation and min-
using the first assumption (constant H vap ) and integrating imum data used. Heat of vaporization is calculated as
vap
the above equation we get follows: H = RB = 8.314 × 4745.66 = 39455.4 J/mol =
39455.4/18 = 2192 kJ/kg. This value gives an error of +4.3%.
B
(6.101) ln P sat = A − Obviously more accurate method of estimation of heat of va-
T porization is through H vap = H sat,vap − H sat,liq , where H sat,vap
where T is absolute temperature and A and B are two positive and H sat,liq can be calculated through generalized correla-
constants specific for each pure substance. This equation sug- tions. Empirical methods of calculation of heat of vaporiza-
gests that ln P vap versus 1/T is a straight line with slope of −B. tion are given in Chapter 7.
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